For example, does $\frac{dy}{dx}\equiv c\mod n$ hold any meaning? How are the rules of calculus modified to allow for this, if possible.
For a simple example, $ax\equiv b\mod n$ behaves the same as $ax=b$ for $x\in[0,n)$, after which there is a discontinuity and then the initial interval output repeats because $ax\equiv((a\mod n)(x\mod n))\mod n$. We could therefore define $\frac{dy}{dx}\equiv a\mod n$ at all points not a multiple of $n$ for lines. It would seem apt to be able to extend this definition to at least higher order polynomials.
One may also need to keep in mind $x\mod n=\frac{x}{n}-n\lfloor{\frac{x}{n}\rfloor}$ as a real number extension for issues of continuity, although this makes derivatives even more confuddling.